WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

3.5 Plate structures 27
a) b)
Figure 3.10 Minimization of the dynamic response of a simply supported plate subjected
to a harmonic point load in the plate center. a) Optimized material distribution (black:
steel, light gray: polycarbonate), b) Frequency responses for the optimized and initial
(homogeneous) plate. The arrow in the frequency response indicates the optimization
frequency. From paper [8].
plates can be optimized in order to quench vibrations and create advanced waveguiding
structures.
Fig.3.10ashows anexample oftheoptimized distributionoftwo elasticmaterials
in a square plate of dimensions 30 × 30cm with a thickness of 3mm. The two
materials are polycarbonate and steel and a maximum of 25% of the material is
allowed to be steel. The plate is simply supported along the rim and is subjected to
a transverse harmonic load in the center. The optimized structure is seen to have a
periodic-like distribution of steel inclusions near the rim of the plate combined with
a large centrally placed circular inclusion. Fig. 3.10b illustrates the corresponding
frequency response plot. It is seen how the vibration level (measured as the total
kinetic energy of the plate) has been significantly reduced compared to the initial
design – a homogeneous plate of equivalent mass.
The optimization algorithm can also be used to design structures with more
advanced functionalities, as presented in Fig. 3.11a. Instead of minimizing the response,
aplateisdesignedthattransportswaveenergyinacircularpathandthereby
creating a ring wave device 4 . The wave is initiated by exciting the plate with two
out-of-phase harmonic forces.
ThewavevectorplotinFig.3.11cshowsthatthedesiredfunctionalityisobtained
with reasonable success. The main part of the wave is guided anti-clockwise through
the designated regions. The optimization setup could be refined by adding more
energy evaluation pointsinthe domainandconsequently a better performance could
4 Ring wave devices have several applications such as in small ultrasonic motors for cameras.